A radio frequency (RF) receiver can downconvert a modulated signal at a carrier frequency to a downconverted signal that is at a baseband frequency. During the downconversion of the modulated signal, certain receiver impairments are introduced due to a non-ideal behavior caused by component process-voltage-temperature (PVT) variations. The receiver impairments degrade system performance. The receiver impairments can be minimized via signal processing. Receiver impairments introduced by a homodyne (direct-conversion) receiver include carrier frequency offset (CFO), IQ mismatch, DC offset, phase noise, etc.
FIG. 1 shows a perfect (or ideal) downconversion circuit 10, operating based on 90° out-of-phase local oscillator signals. The downconversion circuit 10 includes mixers 12, 14 and low pass filters 16, 18. The mixers 12, 14 receive a modulated signal xRF(t) that consists of a baseband signal x(t), which has shifted to a carrier frequency during transmission. The modulated signal xRF(t) may be represented by equation 1 and the baseband signal may be represented by equation 2.xRF(t)=xI(t)cos ωct−xQ(t)sin ωct  (1)x(t)=xI(t)+jxQ(t)  (2)
The mixers 12, 14 receive and multiply the modulated signal xRF(t) by local oscillator signals 2 cos ωct, 2 sin ωct (referred to as “quadrature mixing”) to generate I and Q baseband signal components xI(t), xQ(t) and some unwanted high frequency components. The filters 16, 18 subsequently filter out the unwanted high frequency components from the mixer outputs. Deviation from a 90° difference in phase between the I and Q baseband signal components and difference in gains between the I and Q baseband signal components results in distortion and degrades quality of the resulting baseband signal x(t).
FIG. 2 shows another downconversion circuit 20 operating based on local oscillator signals that are gain and phase mismatched relative to each other. The downconversion circuit 20 includes mixers 22, 24 and low pass filters 26, 28. The mixers 22, 24 receive and multiply a modulated signal xRF(t) by local oscillator signals
      2    ⁢          (              1        +                  ɛ          2                    )        ⁢          cos      ⁡              (                                            ω              c                        ⁢            t                    +                      θ            2                          )              ,            -      2        ⁢          (              1        -                  ɛ          2                    )        ⁢          sin      ⁡              (                                            ω              c                        ⁢            t                    -                      θ            2                          )            that are gain and phase mismatched causing a non-ideal downconversion. The non-ideal downconversion results in mixing of I and Q components of the corresponding baseband signal components to provide received signal components with wI(t) and wQ(t) having IQ mismatch. The received signal components with wI(t) and wQ(t) may be represented by equations 3 and 4, where ε and θ are respectively gain and phase mismatch parameters.
                                          w            I                    ⁡                      (            t            )                          =                              (                          1              +                              ɛ                2                                      )                    ⁢                      (                                                                                x                    I                                    ⁡                                      (                    t                    )                                                  ⁢                cos                ⁢                                  θ                  2                                            +                                                                    x                    Q                                    ⁡                                      (                    t                    )                                                  ⁢                sin                ⁢                                  θ                  2                                                      )                                              (        3        )                                                      w            Q                    ⁡                      (            t            )                          =                              (                          1              -                              ɛ                2                                      )                    ⁢                      (                                                                                x                    I                                    ⁡                                      (                    t                    )                                                  ⁢                sin                ⁢                                  θ                  2                                            +                                                                    x                    Q                                    ⁡                                      (                    t                    )                                                  ⁢                cos                ⁢                                  θ                  2                                                      )                                              (        4        )            Equations 3 and 4 can be represented in a complex form, as shown by equation 5, where equation 6 provides the received signal w(t) with IQ mismatch, equation 7 provides the baseband equivalent received signal prior to IQ mismatch x(t), and equation 8 provides a corresponding complex IQ mismatch parameter a.
                              w          ⁡                      (            t            )                          =                                                            (                                                      cos                    ⁢                                          θ                      2                                                        -                                      j                    ⁢                                          ɛ                      2                                        ⁢                    sin                    ⁢                                          θ                      2                                                                      )                            ⁢                              x                ⁡                                  (                  t                  )                                                      +                                          (                                                                            ɛ                      2                                        ⁢                    cos                    ⁢                                          θ                      2                                                        +                                      j                    ⁢                                                                                  ⁢                    sin                    ⁢                                          θ                      2                                                                      )                            ⁢                                                x                  *                                ⁡                                  (                  t                  )                                                              ≈                                    x              ⁡                              (                t                )                                      +                                          a                ⁡                                  (                  m                  )                                            ·                                                x                  *                                ⁡                                  (                  t                  )                                                                                        (        5        )            w(t)=wI(t)+jwQ(t)  (6)x(t)=xI(t)+jxQ(t)  (7)
                    a        =                              ɛ            2                    +                      j            ⁢                          θ              2                                                          (        8        )            
A final approximation in equation 5 for the received signal w(t) may be obtained by assuming that the gain and phase mismatch parameters ε, θ are small values. In the frequency domain, the received signal w(t) may be represented by equation 9, which shows that IQ mismatch introduces an image a·X*(−ƒ) into the corresponding signal spectrum.W(ƒ)=X(ƒ)+a·X*(−ƒ)  (9)The image a·X*(−ƒ) is a frequency byproduct of the actual signal X(ƒ).